Final answer:
The Drina River is the river that flows along the border with Bosnia and then enters Serbia, which is a tributary of the Sava River and significant in the Balkan region's geography and history.
Step-by-step explanation:
The river that flows along the border with Bosnia and then enters Serbia is the Drina River. The Drina is a tributary of the Sava River which in turn flows into the Danube, one of the major rivers in Europe. The Danube flows eastward, passing through major cities like Vienna, Budapest, and Belgrade, and eventually empties into the Black Sea. It is important to note that the Danube itself does not serve as a border river between Bosnia and Herzegovina and Serbia, but it does flow through Serbia after the confluence with the Sava River. The Drina River's relevance increased during the conflict in Bosnia, as it marked part of the boundary between regions of ethnic division. Today, the Drina River remains significant in defining the border and as a symbol within the Balkan region's complex geography and history.